Could someone explain me how to properly define the intensity of jump size in the Merton jumo diffusion model?I mean, the intensity should be the mean number of arrival of jumps in a time intetval. What are typical values for this intensity? If in the calibration I got a jump intensity of 4, it means that on average I expect 4 jumps in the interval considered, is taht right?Thank you in advance.Best regards

The intensity lambda has the dimensions 1/time.It is indeed the mean number of jumps per time interval.If you adopt annual units, then lambda = 4/year means the expected number of jumps per year is 4.One complication is that there are two expectations in finance: the real-world and the risk-adjusted.If you fit Merton's jump-diffusion process to a time series like the SPX, you will estimate a historical real-world lambda.If you fit Merton's jump-diffusion process to an SPX option chain, you will estimate the risk-adjusted lambda -- this will usually be much higher because it reflects risk-aversion.Let's use annual units.In normal markets for the SPX, you might estimate lambda(real world) ~ 0.1, which means 0.1 jumps per year on average; i.e. one jump every 10 years on average.The corresponding lambda(risk-adjusted) might be ~ 0.25, or one jump expected every 4 years.Conditional on a jump occuring, the mean jump size(risk-adjusted) might be ~ -0.15 or -15%.I haven't done a calibration using Oct 2008 SPX data, and really have little idea what you mightget. I would expect much higher values with perhaps smaller mean sizes. But there isinteraction with the diffusion coefficient (say ~ around 60% annualized for Oct), so hard to say the result. There is a paper by Bates on crash fears in SPX which I recall had some charts of lambda(risk-adjusted) fromoption chain calibrations, plotted over time.